PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 8, Pages 2417–2422 S 0002-9939(99)04851-0 Article electronically published on April 9, 1999
EXACT ESTIMATES FOR INTEGRALS INVOLVING DIRICHLET SERIES WITH NONNEGATIVE COEFFICIENTS
FERENC MORICZ´
(Communicated by Frederick W. Gehring)
Abstract. We consider the Dirichlet series
∞ 1 x akk− − =: f(x),x>0, kX=2 with coefficients a 0 for all k. Among others, we prove exact estimates of k ≥ certain weighted Lp-norms of f on the unit interval (0, 1) for any 0
1. Introduction We shall consider the Dirichlet series with nonnegative coefficients
∞ 1 x (1.1) a k− − =: f(x),a 0, k k≥ kX=2 assuming that the series on the left-hand side converges for all x>0. In Section 3, we shall give a necessary and sufficient condition to ensure this convergence (see (3.9) there). There is a definite interest in the literature (see, for example, [1], [2], [4], and [6]) to estimate certain weighted integrals of φ(f(x)) over the unit interval (0, 1), or over the half real axis (0, ), where φ and the weight functions in question are specified as follows. ∞ Given p q>0, we shall write φ ∆(p, q)ifφ(t) is a nonnegative function ≥ p ∈ q defined on [0, ), φ(0) = 0, φ(t)t− is nonincreasing, and φ(t)t− is nondecreasing ∞ Received by the editors November 19, 1997. 1991 Mathematics Subject Classification. Primary 30B50; Secondary 30B10. Key words and phrases. Dirichlet series, power series, Cauchy-Hadamard criterion for Dirich- let series, line of convergence, Lp-behavior, weight function, slowly decreasing function, Cauchy condensation principle. This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant T 016 393.
c 1999 American Mathematical Society
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on (0, ). We write φ ∆ifφ ∆(p, q)forsomep q>0, but the values of p and q ∞are unimportant∈ in the question∈ concerned. ≥ Clearly, if φ ∆, then φ(t) is nondecreasing on [0, ). Even more is true: If φ ∆(p, q)forsome∈ p q>0, then ∞ ∈ ≥ (1.2) vpφ(t) φ(vt) vqφ(t)for0 v 1andt 0. ≤ ≤ ≤≤ ≥ The weight functions in this paper are “slowly decreasing.” More precisely, we shall write λ Λifλ(t) is a positive, nonincreasing function on [1, )andsuch that ∈ ∞ λ(2t) λ(2t) 0 < lim inf lim sup < 1. t λ(t) ≤ t λ(t) →∞ →∞
2. New results We shall use the following notations: (2.1) I := 2n, 2n +1,... ,2n+1 1 ,n=1,2,... ; n { − } n n (2.2) αn := 2− ak,An:= αm,n=1,2,... ; k I m=1 X∈ n X where ak : k =2,3,... is the sequence of coefficients in (1.1). Our{ main result is the} following Theorem 1. If λ Λ, Φ ∆,andf(x)is defined in (1.1), then there exists a positive constant K∈depending∈ only on φ and λ such that
∞ 1 n (2.3) K− λ(2 )φ(A2n+1 1) − n=0 X 1 x 1 x 1 λ((1 2− )− )(1 2− )− φ(f(x))dx ≤ − − Z0 ∞ K λ(2n)φ α . ≤ k n=0 k I X X∈ n
It is plain that estimate (2.3) remains valid if A2n+1 1 is replaced by k I αk − ∈ n on the left-most side and/or if α is replaced by A n+1 on the right-most k In k 2 1 side of (2.3). ∈ − P First, we consider the specialP case, where
γ λ(t):=t− ,γ>0. Making use of the Cauchy condensation principle yields the following Corollary 1. If γ>0and φ ∆, then there exists a positive constant K depending only on γ and φ such that ∈
∞ 1 ∞ 1 φ(An) x γ 1 φ(An) (2.4) K− (1 2− ) − φ(f(x))dx K . nγ+1 ≤ − ≤ nγ+1 n=1 0 n=1 X Z X Since x x ln 2 1 2− x ln 2 for 0 x 1, 2 ≤ − ≤ ≤ ≤
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from (2.4) it follows immediately that
∞ γ 1 1 φ(An) x − φ(f(x)) L (0, 1) if and only if < . ∈ nγ+1 ∞ n=1 X 1 Second, we consider the even more special case, where γ := 1, i.e., let λ(t):=t− . Corollary 2. If φ ∆, then there exists a positive constant K depending only on φ such that ∈ (2.5) 1 1 ∞ φ(An) ∞ ∞ φ(An) K− φ(f(x))dx φ(f(x))dx K . n2 ≤ ≤ ≤ n2 n=1 0 0 n=1 X Z Z X In particular, if φ(t):=tp for some p>0, then (2.5) provides an exact estimate for the Lp-norm of the sum of the Dirichlet series (1.1).
3. Auxiliary results Our estimation is based on the close relationship between Dirichlet series and power series, which enables us to derive exact estimates for integrals involving the former one by making use of exact estimates for integrals involving the latter one, which has been available in the literature. To go into details, consider the power series with nonnegative coefficients
∞ (3.1) b xk =: g(x),b 0, k k≥ kX=1 assuming that the series on the left-hand side converges for all 0 x<1. According to the Cauchy-Hadamard criterion, the condition ≤ 1/k (3.2) lim sup bk 1 k | | ≤ →∞ is necessary and sufficient in order that the radius of convergence of series (3.1) be at least 1. We shall use the following notation: n (3.3) βn := bk,sn:= bk, k I k=1 X∈ n X where In is defined in (2.1). The next theorem was proved by Leindler [3]. Theorem 2. If λ Λ, φ ∆,andg(x)is defined in (3.1), then there exists a positive constant K∈depending∈ only on λ and φ such that
∞ 1 n (3.4) K− λ(2 )φ(s2n+1 1) − n=0 X 1 1 1 λ((1 x)− )(1 x)− φ(g(x))dx ≤ − − Z0 ∞ K λ(2n)φ(β ). ≤ n n=0 X
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The first inequality in (3.4) was first proved by Mateljevic and Pavlovic [5] in the γ special case λ(t):=t− for some γ>0. The second inequality was proved also by them with the seemingly greater quantity φ(s2n+1 1) instead of φ(βn). However, it is easy to see that, under the conditions of Theorem− 2, the sums of the series
∞ n ∞ n λ(2 )φ(s2n+1 1)and λ(2 )φ(βn) − n=0 n=0 X X are of the same order of magnitude. Analyzing the proofs in [3] and [5], one can conclude the following improvement of the first inequality in (3.4). Lemma 1. Let n r := 1 2− ,n=0,1,2,... . n − Under the conditions of Theorem 2, we have (3.5) ∞ 1 n K− λ(2 )φ(s2n+1 1) − n=n X0 1 1 1 λ((1 x)− )(1 x)− φ(g(x))dx, n0 =0,1,2,... ; ≤ r − − Z n0 with the same constant K as in (3.4). The next auxiliary result is of crucial importance for our purposes, but it is of special interest in itself. It makes it possible to pass from power series into Dirichlet series. Lemma 2. Given a Dirichlet series (1.1), introduce the power series
∞ n (3.6) αnu =: h(u), n=1 X where αn is defined in (2.2). If the radius of convergence of the series on the left-hand side in (3.6) is at least 1,then 1 x x x (3.7) 2− − h(2− ) f(x) h(2− ) for x > 0. ≤ ≤ Proof. By (1.1), (2.1), and (2.2), it is plain that
∞ 1 x f(x)= akk− − n=1 k I XX∈ n ∞ n( 1 x) n x 2 − − 2 α = h(2− ). ≤ n n=1 X On the other hand,
∞ (n+1)( 1 x) n f(x) 2 − − 2 α ≥ n n=1 X 1 ∞ (n+1)x 1 x x =2− αn2− =2− − h(2− ). n=1 X
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Combining Lemma 2 with the Cauchy-Hadamard criterion, while making use of the obvious inequality 1 a α k α , 2 n ≤ k ≤ n k I X∈ n yields the following
Lemma 3. If a 1/n (3.8) 2x > lim sup k , n k →∞ k In n X∈ o then the Dirichlet series (1.1) converges. If the inequality sign is reversed in (3.8), then series (1.1) diverges. In particular, the Dirichlet series (1.1) converges for all x>0if and only if a 1/n (3.9) lim sup k 1. n k ≤ →∞ k In n X∈ o As a by-product, we obtain an analogue of the Cauchy-Hadamard criterion of (absolute) convergence of the more general Dirichlet series
∞ z (3.10) ckk− ,z:= x + iy, kX=2 where ck is a sequence of complex numbers. Denote{ } by Θ the supremum of those x := Re z, for which series (3.10) converges. By what we have discussed above, it is clear that Θ is determined by the equation
1/n Θ 1 ck 2 − = lim sup | | . n k →∞ k In n X∈ o (In (1.1) we have z := 1 + x.) Furthermore, the line given by the equation z =Θ can be viewed as the “line of absolute convergence,” which means that series (3.10) converges absolutely for all z with x := Re z>Θ, while it diverges for all z := x< Θ.
4. Proofs of Theorem 1 and Corollaries 1 and 2 Proof of Theorem 1. (i) By Lemma 2, Theorem 2, and (2.2), while introducing x u := 2− ,wehave
1 x 1 x 1 I:= λ((1 2− )− )(1 2− )− φ(f(x))dx 0 − − Z 1 1 1 du λ((1 u)− )(1 u)− φ(h(u)) ≤ − − u ln 2 Z1/2 2K ∞ λ(2n)φ α . ≤ ln 2 k n=0 k I X X∈ n This proves the second inequality in (2.3).
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(ii) By Lemmas 1 and 2, (1.2), and (2.2), we get similarly that 1 1 1 1 du I λ((1 u)− )(1 u)− φ(2− uh(u)) ≥ − − u ln 2 Z1/2 1 K− 1 ∞ n min 1, λ(2 )φ(A2n+1 1). ≥ 2p ln 2 2p 1 − − n=1 n o X Hence the first inequality in (2.3) follows easily. Proof of Corollary 1. From (2.3) it follows immediately that 1 1 ∞ nγ x γ 1 K− 2− φ(A2n+1 1) (1 2− ) − φ(f(x))dx − ≤ − n=0 0 X Z ∞ nγ K 2− φ(A2n+1 1). ≤ − n=0 X Making use of the Cauchy condensation principle yields
∞ nγ ∞ φ(Ak) ∞ φ(Ak) 2− φ(A2n+1 1) = . − ≥ kγ+1 kγ+1 n=0 n=0 k I k=1 X X X∈ n X This proves the first inequality in (2.4). The second inequality in (2.4) can be proved analogously. Proof of Corollary 2. It follows immediately from Corollary 1 when γ = 1, except for the right-most inequality in (2.5). To prove it, let us observe that j f(x + j) 2− f(x)forx>0andj=1,2,... . ≤ Thus, by (1.2), we have 1 1 ∞ ∞ j q 1 φ(f(x))dx φ(2− f(x))dx (1 2− )− φ(f(x))dx. ≤ ≤ − 0 j=0 0 0 Z X Z Z This proves the right-most inequality in (2.5). References
1. L. Leindler, Generalizations of some theorems of Mulholland concerning Dirichlet series, Acta Sci. Math. (Szeged) 57(1993), 401-418. MR 94j:41010 2. L. Leindler, Improvements of some theorems of Mulholland concerning Dirichlet series, Acta Sci. Math. (Szeged), 58(1993), 281-297. MR 95g:40002 3. L. Leindler, On power series with positive coefficients, Analysis Math. 20 (1994), 205-211. MR 95k:40010 4. L. Leindler and A. Meir, Inequalities concerning Dirichlet series and integrals, Acta Sci. Math. (Szeged) 59(1994), 209-220. MR 95h:26025 5. M. Mateljevic and M. Pavlovic, Lp-behavior of power series with positive coefficients and Hardy spaces, Proc. Amer. Math. Soc. 87 (1983), 309-316. MR 84g:30034 6. H. P. Mulholland, Some theorems on Dirichlet series with positive coefficients and related integrals, Proc. London Math. Soc. 29(1929), 281-292.
Bolyai Institute, University of Szeged, Aradi Vertanuk Tere 1, 6720 Szeged, Hungary E-mail address: [email protected]
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